Optimal Numeraires for Risk Measures∗

Can the usage of a risky numeraire with a greater than risk free expected return reduce the capital requirements in a solvency test? I will show that this is not the case. In fact, under a reasonable technical condition, there exists no optimal numeraire which yields smaller capital requirements than any other numeraire. 1 Statement and Proof of the Result Can the usage of a risky numeraire with a greater than risk free expected return reduce the capital requirements in a solvency test? I will show that this is not the case. In fact, under a reasonable technical condition, there exists no optimal numeraire which yields smaller capital requirements than any other numeraire. We consider a one period setup. Terminal nominal values are modelled as essentially bounded random variables X ∈ L∞ on some probability space (Ω,F ,P). Random variables that coincide almost surely are identified. The riskiness of a portfolio is quantified by a convex risk measure ρ on L∞ satisfying the following “coherence” axioms (introduced by Artzner et al. [1] and further extended to the convex case by Föllmer and Schied [5, 6]): convexity: ρ(λX + (1− λ)Y ) ≤ λρ(X) + (1− λ)ρ(Y ) for λ ∈ [0, 1], (1) monotonicity: ρ(X) ≥ ρ(Y ) if X ≤ Y , (2) cash-invariance: ρ(X +m) = ρ(X)−m for m ∈ R, (3) normality: ρ(0) = 0. (4) It is legitimate practice to discount terminal values by a numeraire — one euro tomorrow is less than one euro today. We denote by r ≥ 0 the prevailing risk free rate. The regulatory required capital (the “solvency capital ∗I wrote this paper during my appointment as Visiting Professor in the Faculty of Business at the University of Technology in Sydney. I gratefully acknowledge the hospitality of Eckhard Platen. I thank Michael Kupper and Gregor Svindland for helpful comments.